Recall now that in non-archimedean geometry, the tropicalization of a toroidal embedding of Artin stacks \((\mathcal {U}\subset \mathcal {X})\)  (or more generally a logarithmic algebraic stack) is meant to be a generalized cone complex \(\Sigma (\mathcal {X})\) , and its natural compactification \(\overline{\Sigma }(\mathcal {X})\)  which is a generalized extended cone complex, together with a surjective proper continuos map (that we baptise functorial analytic tropicalization) \(\operatorname{trop}_{\mathcal {X}}^{an}:|\mathcal {X}^{\beth }|\rightarrow \overline{\Sigma }(\mathcal {X})\)  which is functorial with respect to locally toric morphisms, see [1]} and [2]}.
